Lecture 19.pptx Graph Theory PDF

Ex: Find all possible subgraph for the following G graph e3 e1 V1 ᴏ ᴏ V2 e2 deg (V1)= , deg (V2)= , deg (G)=  e2 e1 Aᴏ ᴏB...

Ex: Find all possible subgraph for the following G graph e3 e1 V1 ᴏ ᴏ V2 e2 deg (V1)= , deg (V2)= , deg (G)=  e2 e1 Aᴏ ᴏB e3 Fᴏ e4 e6 ᴏC Eᴏ e7 e5 ᴏD A walk from D to A can be written as following De5Ce3 Be1 A Write the following walks E to B: A to E: A to A: C to F: Def: let G be a graph of n-vertices then: 1- A walk from V to W is a finite sequence of vertices and edges of the form: V0 e1V1e2V2...enVn where V0=V and Vn=W 2- The trivial walk on V consists only V. V0 e1V1e2V2...enVn ei e j i, j 3- A path from V to W is a walk with no repeated edges. V0 e1V i.e. 1e2V2...enVn Vi V j i, j s.t. 4- A simple path, is a path with no repeated vertices. i.e. s.t. 5- A closed walk is a walk that starts and ends at same vertex. 6- A circuit is a closed walk which has no repeated edge. Ex: e4 e2 V1ᴏ e1 ᴏ ᴏ V2 V3 e3 V1e1V2 e3V3 is a walk from to is it path? V1e1V2 e2V3e4V3e2V2 is it walk? Is it path? write a path from V3 to V1 write a simple path from V1 to V3 write a walk that is not path from V1 to V3 write a walk that is path but not simple path from V 2 to V3 write a close walk from V2 write a circuit from V3 Ex: Draw the graph for the following walk Ae1Ce2 Be5 Ee4Ce3 D Write 5 different close walk from C. Def: A simple circuit is a circuit with no repeated vertices. Ex: determine path, simple path, close, circuit or simple circuit. e4 e5 e3 ᴏ V3 ᴏ V4 e2 e6 e7 e10 V1 ᴏ ᴏ V2 e1 e8 ᴏ V5 ᴏ V6 e9 V1e1V2 e3V3e4V3e5V4 V1e1V2 e3V3e5V4 e5V3e6V5 Path: no repeated edge Simple path: no repeated vertex V2V3V4V5V3V6V2 Close: start and end with same vertex V2V3V4V5V6V2 Circuit: no repeated edge Simple circuit: no repeated vertex V2V3V4V5V6V3V2 V1   Def: let G be a graph. Two vertices V and W are connected, if a walk  between V and W. the graph G is called connected, if for any two vertices, a walk.  V , W  V (G ) i.e. G connected a walk between V and W. Ex: For the following G, determine ᴏV4 the connectivity V2 ᴏ ᴏ V3 ᴏV5 V1 ᴏ V6 ᴏ V2 and V3 V2 and V6 The graph G Ex: determine the connectivity V4 V3 V1 and V3 G A B V1 V2 V5 F ᴏH C E D B D A F C E Def: Let G be a graph an Euler circuit for G is a circuit which contain all vertices and every edge (only once). i.e. Euler circuit 1- Circuit (close not repeated edge) 2- Use every vertex at least once 3- Use every edge exactly one time. Ex: is it an Euler circuit? ᴏV4 V2 ᴏ ᴏ V3 ᴏV5 V1 ᴏ V6 ᴏ Ex: Konigsberg bridges problem. A ᴏ Dᴏ ᴏB ᴏ C Theorem: If G is a graph having an Euler circuit, then every vertex is of even degree. ᴏV4 Ex: Does G have en Euler circuit? V2 ᴏ ᴏ V3 ᴏV5 V1 ᴏ V6 ᴏ A B Ex: Does G have en Euler circuit? D C Does every G contains even degree for all vertices an Euler circuit? Theorem: If G is connected and the degrees of all vertices are even, then G ha san Euler circuit.  i.e. Euler circuit  i) is even G connected + deg(V i. Ex: check the Euler circuit A D E I B C J H F K  Matrix representation of a graph Def: Let G be a graph of n-vertices. The matrix adjacency of G (or adjacent matrix of G) is a matrix A of asize ij n×n whose entries ai j : is the number of edges connected vertices V and V i j ai j could be either 0 or positive integers. Ex: B F I A K C D H A B C D F H 0 I K    1  1    1  A   0  0    0  0    Ex: Let 1 2 0 1   2 0 0 1 A  1 1 1 0   1 2 0 1   Draw a graph G having A as its adjacent matrix.  ai a j Remark: the adjacent matrix of a graph G is symmetric j i as Ex: Draw the graph G having 1 2  A    2 1 Ex: Draw the graph G having 1 2 1   A  2 1 3 1 2 1    I 33 Ex: Draw the identity matric